Since you say you only need this up to 100, I don't see why you can't just compute it exactly using a system with proper bignums.
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David EppsteinOct 29 '11 at 7:28

David, I didn't want to deal with numbers like 2^10000 first but after learning that it is not possible to get a closed formula for the exact values, I decided to use the GMP (GNU Multiple Precision) Bignum Library. Thanks everybody.
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NickOct 29 '11 at 19:27

Essentially, the approximations are geometric series starting with the dominant term in your sums, and except for k near n/2, should serve you well, especially if you collapse adjacent terms in the form of n^2 +1 choose j. For k near n/2, Michael Lugo has some suggestions in his answer to the question above. If you do a search on "binomial coefficients sum", you will find other MathOverflow posts considering sums similar to yours which might help you.

I think getting a nice closed form for the exact values for your numerators in general is unsolvable.